Recently, due to its numerous applications, the spectra of the bounded operators over Banach spaces have been studied extensively. This work aims to collect some of the investigations on the spectra of difference operators or matrices on the Banach space c in the literature and provide a foundation for related problems. To the best of our investigations, the problem has been solved over the sequence space c so far up to maximum order 2. In the present work, the fine spectra of the difference operator \(\Delta ^m, m\in \mathbb {N}\) on c have been computed. The generalized difference operator \(\Delta ^m\) on the Banach space c is defined by \((\Delta ^mx)_k= \sum _{i=0}^m(-1)^i\left( {\begin{array}{c}m\\ i\end{array}}\right) x_{k-i},\;k=0,1,2,3,\dots \) with \( x_{k} = 0\) for \(k<0\). Indeed, the operator \(\Delta ^m\) is represented by an \((m+1)\)-th band matrix which generalizes several difference operators such as \(\Delta ,\Delta ^2,B(r,s)\) and B(r, s, t) etc, under different limiting conditions. Initially, we provide some essential background results on the linearity and boundedness of the backward difference operator \(\Delta ^m\). Finally, the sets for the spectrum and fine spectra such as the point spectrum, the continuous spectrum and the residual spectrum of the defined operator on the space c have been computed. The geometrical interpretation for the spectral subdivisions of the above difference operator is also incorporated.

最近，由于其众多的应用，对Banach空间上的有界算子的光谱进行了广泛的研究。这项工作旨在收集有关文献中Banach空间c上差分算子或矩阵谱的一些研究，并为相关问题提供基础。尽我们所能研究的结果，该问题已经解决了序列空间c上的最大阶数2。在本工作中，差分算子\（\ Delta ^ m，m \ in \ mathbb {已计算出c上的N} \）。Banach空间c上的广义差分算子\（\ Delta ^ m \）由下式定义\（（\ Delta ^ mx）_k = \ sum _ {i = 0} ^ m（-1）^ i \ left（{\ begin {array} {c} m \\ i \ end {array}} \ right ）x_ {ki}，\; k = 0,1,2,3，\ dots \），其中\（x_ {k} = 0 \）对于\（k <0 \）。确实，算子\（\ Delta ^ m \）由第（\（（m + 1）\）个频带矩阵表示，该矩阵概括了几个差分算子，例如\（\ Delta，\ Delta ^ 2，B（r， s）\）和B（r，  s，  t）等，在不同的限制条件下。最初，我们提供了一些关于后向差分算子\（\ Delta ^ m \）的线性和有界性的基本背景结果。最后，计算了空间c上定义算子的光谱和精细光谱的集合，例如点光谱，连续光谱和残留光谱。还合并了上述差分算子的频谱细分的几何解释。